Violation of the two-time Leggett-Garg inequalities for a harmonic oscillator

We investigate the violation of the Leggett-Garg inequalities for a harmonic oscillator in various quantum states and with various choices of a projection operator for a dichotomic variable. We focus on the two-time quasiprobability distribution function with a dichotomic variable constructed with the position or momentum operator of a harmonic oscillator. Our results are the generalization of the previous work by Mawby and Halliwell [Phys. Rev. A 107, 032216 (2023)], but the new points are the following. We first obtain the explicit expression for the two-time quasiprobability distribution function for the (thermal) squeezed coherent state. Second, we find the two-time quasiprobability distribution function with the dichotomic variable and the projection operator constructed in terms of the momentum operator. Third, we demonstrate that the violation of the Leggett-Garg inequalities can be boosted by adopting the dichotomic variable and the projection operator defined by a finite range of the position/momentum, in which a larger violation appears for the ground state and the squeezed state. We give an intuitive interpretation when the violation of the Leggett-Garg inequalities appears. We also present a mathematical formula to compute the quasiprobability distribution function by using the integral representation of the Heaviside function, which is useful to generalize it to a quantum field theory.